Manifold & Metric: Does it Need a Metric?

In summary, a manifold does not necessarily have a metric, as it may not have a global metric due to the lack of a "distance" between any two points. The hierarchy of manifolds includes a set, a topology, a topological manifold, and a differential manifold. Non-metrizable manifolds, such as the Prüfer manifold, do exist. The notions of "metric space" and "metric tensor" are related but not the same, and Lorentzian metrics do not get their topology from the bundled indefinite bilinear form.
  • #1
princeton118
33
0
Does a manifold necessarily have a metric?
Does a manifold without metric exist? If it exists, what is its name?
 
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  • #2
Since a manifold is locally Euclidean, it must always have a local metric. However, it does not follow that there will be a "distance" between ANY two points and so there may not be a "global" metric.
 
  • #3
the hierarchy is something like this. You take a set. Add a topology. It becomes a tpological space. Add an atlas, it becomes a topological manifold. Change for a differentiable atlas. It becomes a differential manifold. Add a Riemannian or Pseudo-Riemannian of Lorentzian or whatever metric and it becomes a Riemannian (resp. Pseudo-Riemannian, Lorentzian, whatever) manifold.
 
  • #4
Topological manifolds (sets with a topology locally homeomorphic to Rn) do not necessarily admit a metric. There are then many non-metrizable manifolds, such as the Prüfer manifold. Urysohn's metrization theorem will let you know if your manifold is metrizable.
 
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  • #5
Ditto slider142 about topological manifolds. I have a hunch the OP will need to know that "metric space" is not the same thing as the notion of "metric tensor" from Riemannian geometry, although they are certainly related. But "metric tensor" from Lorentzian geometry is not much like "metric" from "metric space"!

About smooth manifolds: you can give a smooth manifold additional structure, perhaps by defining a Riemannian or Lorentzian metric tensor. As Halls hinted, as per the fundamental local versus global distinction in manifold theory, even after defining a Riemannian or Lorentzian metric tensor, there will be multiple distinct notions of "distance in the large" which may or not correspond roughly to the notion of "metric" fro m "metric space". In particular, Lorentzian metrics get their topology from the (locally euclidean) topological manifold structure, not from the bundled indefinite bilinear form.

(I'm being a bit more sloppy than usual due to PF sluggishness.)
 

Related to Manifold & Metric: Does it Need a Metric?

1. What is a manifold?

A manifold is a mathematical space that is smooth and locally resembles Euclidean space. It can be described as a set of points that can be mapped onto a subset of Euclidean space using a set of coordinate charts.

2. What is a metric?

In mathematics, a metric is a function that measures the distance between two points in a given space. In the context of manifolds, a metric is used to define the notion of distance and angles, which are crucial for understanding the geometry of the manifold.

3. Why do we need a metric on a manifold?

A metric is necessary on a manifold because it allows us to define and measure distances between points on the manifold. This is important for understanding the geometry of the manifold and allows us to perform calculations and make predictions based on this geometry.

4. Can a manifold exist without a metric?

Yes, a manifold can exist without a metric. In fact, there are many types of manifolds that do not have a natural metric defined on them. However, in order to understand the geometry of a manifold, it is often helpful to define a metric on it.

5. What is the relationship between a manifold and its metric?

The metric of a manifold is closely related to its geometry. It allows us to define the length of curves and calculate angles between vectors on the manifold. Additionally, the metric can be used to define other important geometric concepts such as curvature. In summary, a metric is a fundamental tool for understanding the geometry of a manifold.

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